Traditionally, moment series are solved recursively upward in \(k\) :
\({\Phi }_0 \equiv 1\)  by normalization of probability, \(R_0 \equiv {\Phi }_1\)  is estimated by the mean-field equations, and so forth. The
lower-order moments are assumed to be not only the larger components
of the variation, but also those components most robust in a simple
model against parameter imprecision or mis-specification of model
details [1]}. Therefore, as emphasized
in [2]}, [3]}, it is
counterintuitive that the recursive solution for \({\hat{R}}_k\)  is
anchored not at \(k \rightarrow 0\)  by \(x\)  in the mean-field solution,
but rather at \(k \rightarrow \infty \)  by ratios of parameters in
\(\mathcal {L}\)  that generally differ from the mean of \(x\)  at any of its
fixed points, and which are related to probability currents that can
arise in networks of nonzero deficiency
(see [4]}, [5]} for
elaborations on deficiency). The downward-going recursion is stable
for \(k / \left< {\rm n}\right>\)  sufficiently large, so the asymptotic
value can be imposed on \(R_k\)  at large finite \(k\) , and truncation
errors are exponentially attenuated as \(k\)  decreases. However, the
downward recursion becomes unstable below some characteristic scale
for \(k / \left< {\rm n}\right>\) , and errors that were initially
attenuated are amplified, along with numerical precision errors, in a
diverging asymptotic expansion that cannot be continued if \(n_C\)  is
large enough.
